# Ring and Module Theory Qual Review

# Robert Won Prof. Rogalski

## 1 (Some) qual problems

• (Spring 2007, 2) Let $I , J$ be two ideals in a commutative ring R (with unit).

(a) Define $K = \{ r : r J \leq I \}$ . Show that K is an ideal

(b) If R is a PID, so $I = \langle i \rangle , J = \langle j \rangle$ , give a formula for a generator k of K.

• (Spring 2007, 3) Describe up to isomorphism all the $\mathbb { R } [ x ] .$ -module structure one might put on a 3 dimensional real vector space (extending the R action).

Fundamental theorem of modules over a PID.

• (Fall 2009, 8) Let R be a commutative ring with identity. Suppose I and J are ideals of R such that $R / I$ and $R / J$ are noetherian rings. Prove that $R / ( I \cap J )$ is also a noetherian ring.

• (Spring 2012, 3) Let R be a commutative ring.

(a) Suppose that R is noetherian. Show that if $\varphi : R  R$ is a surjective ring homomorphism, then it is injective.

(b) If R is not noetherian, must a surjective ring homomorphism be injective? Prove or give a counterexample.

item (Spring 2012, 5) Consider $f \in F [ x ]$ where F is an algebraically closed field. Suppose that f has the property that for all matrices $A \in M _ { n } ( F )$ of any size n, if $f ( A ) = 0$ , then A is a diagonalizable matrix; then we say the polynomial f forces diagonalizability.

(a) Characterize a simple rule exactly which polynomials in $F [ x ]$ prove diagonalizability. Prove your answer.

(b) Fix $m \geq 1$ . Is every square matrix A with entries in F satisfying $A ^ { m } = I$ diagonalizable? (The answer depends on F ).

• (Fall 2009, 2) Let R denote a commutative ring and I an ideal, $I \neq R .$

(a) Give an example where $R / I$ has nilpotents but R doesn’t.

(b) Give an example where R has nilpotents but $R / I$ doesn’t.

• (Fall 2009, 3) Let $\varphi : \mathbb { C } [ x ] \to F$ be a ring homomorphism where F is a field, $\varphi ( 1 ) \neq 0$

(a) Give an example where ϕ is not onto.

(b) If $\varphi$ is onto, show that $F \cong \mathbb { C }$

• (Fall 2009, 4)

(a) Give an example of two finitely generated Z-modules, M and N such that $M , N$ are not isomorphic (as Z-modules) but $\mathbb { Q } \otimes _ { \mathbb { Z } } M \cong \mathbb { Q } \otimes _ { \mathbb { Z } } N$ (as Q-modules).

(b) Let M be a finitely generated R[x]-module, described using the classification of $\mathrm { f . g . }$ modules over a PID. Give a similar description of $\mathbb { C } [ x ] \otimes _ { \mathbb { R } [ x ] } M$ as a C[x]-module.

## 2 (Some) ring things to know

• Basic facts and definitions (homomorphisms, isomorphism theorems, subrings, ideals, quotient rings, etc.)

• Any finite integral domain is a field.

• Isomorphism theorems

• A commutative ring R is a field if and only if its only ideals are 0 and R.

• Kernels of ring homomorphisms are ideals.

• For a commutative ring R and an ideal I, R/I is a domain (resp. field) if and only if I is prime (resp. maximal).

• (Chinese Remainder Theorem) Let $A _ { 1 } , \ldots , A _ { k }$ be ideals of R. Then map

$$
R \to R / A _ { 1 } \times \cdots \times R / A _ { k }
$$

has kernel $A _ { 1 } \cap \cdots \cap A _ { k }$ . If for each $i \neq j , A _ { i }$ and $A _ { j }$ are comaximal, then the map is surjective and $A _ { 1 } \cap \cdot \cdot \cdot A _ { k } = A _ { 1 } \cdot \cdot \cdot A _ { k }$ so

$$
R / ( A _ { 1 } \cdots A _ { k } ) = R / ( A _ { 1 } \cap \cdots \cap A _ { k } ) \cong R / A _ { 1 } \times \cdots \times R / A _ { k } .
$$

• fields $\subsetneq$ Euclidean domains $\subsetneq \mathrm { P I D s } \subsetneq \mathrm { U F D s } \subsetneq$ Integral domains $\subsetneq$ Rings Examples showing strict inclusion: $F [ x ] , \mathbb { Z } [ ( 1 + { \sqrt { - 1 9 } } ) / 2 ] , F [ x , y ] , \mathbb { Z } [ { \sqrt { - 5 } } ] , \mathbb { Z } _ { 4 }$

• A Euclidean domain is a PID, an ideal is generated by an element of minimum norm.

• If $( a , b ) = ( d )$ , where $d = \operatorname* { g c d } ( a , b )$

• If R[x] is a PID then R is a field.

• If F is a field then F [x] is a Euclidean domain.

• Every prime ideal in a PID is maximal.

• In a UFD, an element is prime if and only if it is irreducible.

• R is a UFD if and only if R[x] is a UFD.

• Gaussian integers

• (Gauss’ Lemma) Let R be a UFD with field of fractions F and $p ( x ) \in R [ x ]$ . If $p ( x )$ is reducible in $F [ x ]$ then $p ( x )$ is reducible in $R [ x ]$ .

• (Eisenstein’s criterion) Let P be a prime ideal of an integral domain R and let $f ( x ) =$ $x ^ { n } + a _ { n - 1 } x ^ { n - 1 } + \cdot \cdot \cdot + a _ { x } +$ a0 be a polynomial in $R [ x ]$ . Suppose $a _ { n - 1 } , \dots , a _ { 1 } , a _ { 0 } \in P$ and $a _ { 0 } \notin P ^ { 2 }$ . Then $f ( x )$ is irreducible.

## 3 (Some) module things to know

• Basic facts and definitions (homomorphisms, isomorphism theorems, submodules, quotient ideals, direct products, cyclic modules)

• An F [x]-module V is an F -vector space with a linear transformation $V  V$

• Free modules (Let A be any set and consider the free module $F ( A )$ . If M is any R-module, $\varphi A  M$ any homomorphism, this lifts to a unique R-module homomorphism $\Phi : F ( A ) $ M .)

• Tensor product (commutative, associative, right exact, splits over direct sums)

• (Universal property) Any R-bilinear map $M \times N \to L$ induces a unique R-module homomorphism ${ \cal M } \otimes _ { { \cal R } } { \cal N }  { \cal L }$

• Extension of scalars

• Hom (is left exact)

• Exact sequences (short, split)

• Projective modules (direct summand of a free module, lifting property, every short exact sequence ending with a projective splits)

(Lifting property). Given a surjection $M  N$ and any homomorphism $\varphi P  N , \varphi$ lifts to a (not necessarily unique) homomorphism $P  N$

• Injective modules (lifting property, every short exact sequence starting with an injective splits) (Lifting property dual to the projective property). For any injection $L \ \to \ M$ , and any homomorphism $\varphi L \to Q , \varphi$ lifts to a homomorphism $Q  M$

• (Baer’s criterion) An R-module Q is injective if and only if for every left ideal I, any module homomorphism $I  Q$ can be extended to one $R \to Q$

• If R is a PID, Q is injective if and only if $r Q = Q$ for all $0 \neq r \in R$

• Flat modules (tensoring with a flat module is exact)

• M is a noetherian R-module if nad only if every nonempty set of submodules of M contains a maximal element if and only if every submodule ofM is finitely generated.

• (Fundamental Theorem of Modules over a PID)

Invariant factor form

Let R be a PID and M a finitely gnereated R-module. Then

$$
{ \cal M } \cong R ^ { r } \oplus R / ( a _ { 1 } ) \oplus \cdots \oplus R / ( a _ { m } )
$$

for some $r \in \mathbb { Z } _ { \geq 0 }$ and $a _ { 1 } \mid a _ { 2 } \mid \cdots \mid a _ { m }$

Rational canonical form

Elementary divisor form:

$$
{ \cal M } \cong R ^ { r } \oplus R / ( p _ { 1 } ^ { \alpha _ { 1 } } ) \oplus \cdots R / ( p _ { t } ^ { \alpha _ { t } } )
$$

for some $r \in \mathbb { Z } _ { > 0 }$ and $p _ { 1 } ^ { \alpha _ { 1 } } , \ldots , p _ { t } ^ { \alpha _ { t } }$ are positive powers of not necessarily distinct primes. Jordan canonical form

• Characteristic polynomials, minimal polynomials